Biomedical Engineering Reference
In-Depth Information
11.2 A mixture model for water uptake, degradation and erosion
from polymeric matrices
Following the lines of [43], we develop a general tridimensional model for the degra-
dation of PLA upon water absorption and subsequent hydrolysis. First, we describe
the nature of the model and its governing equations and then complement it with
specific initial and boundary conditions. The main difficulty consists in the defi-
nition of suitable constitutive laws for the model coefficients. The novelty of the
present work consists in the application of atomistic simulation to estimate the most
significant properties of PLA as it degrades and erodes, giving rise to a multiscale
description of the material.
11.2.1 Macroscale governing equations for a polymer mixture
We describe a polydisperse polymeric network as a collection of different linear
chains of repeating units. Each chain is characterized by its degree of polymeriza-
tion,
x
, defined as the number of repeating units. The system is discretized into
N
number of constituents by defining
N
mutually exclusive equidistent partitions
P
i
=[
−
/
,
+
/
[
=
,...,
N
, of length
x
1
of the degree of polymeriza-
tion spectrum. Each partition
P
i
represents the class of chains whose average degree
of polymerization is
x
i
.
Diffusion driven by negative density gradients is the driving force for mass trans-
port. To account for diffusion, we introduce a spatial coordinate
x
characterizing the
location of a particle of the mixture with volume d
V
. At each particle, water, drug,
and
N
polymeric constituents coexist. Since the mascroscale spans over the entire
length of the polymer matrix, external boundaries have to be taken into account. In
this respect, an open system is considered as water penetrates into the polymer matrix
from the outside aqueous environment and polymeric mass is lost to the exterior. The
mass balances for each individual constituent yield the system of reaction-diffusion
equations constituting the mathematical model.
Let
x
i
x
1
2
x
i
x
1
2
,for
i
1
be the partial density of chains of average degree of polymeriza-
tion
x
i
in a representative control volume,
dV
, that corresponds to point
x
at time
t
and let
w
i
=
ρ
i
/
∑
ρ
i
=
ρ
i
(
x
,
t
)
ρ
i
be the weight fraction of chains of length
x
i
.Let
ρ
w
(
x
,
t
)
be the
i
partial density of water. We also denote with
˜
ρ
=
∑
ρ
i
the partial density of all poly-
i
mer sub-fractions and with
ρ
i
the total density of the mixture. Finally,
for the forthcoming description, we denote with
ρ
=
ρ
w
+
∑
i
the vector of
partial densities of each component that completely identifies the state of the mix-
ture, i.e. this is the
state space
of the macroscale model. For notation cleanliness, we
shall omit the dependence on space and time (
x
and
t
) of partial densities.
Mass can be neither created nor destroyed, but in the present case, polymer chains
and water can diffuse in/out of d
V
through its boundary. Moreover, polymeric con-
stituents interconvert from one to another due to scission reactions. The mass balance
of each constituent in d
V
states that the time rate of change of mass existing in d
V
is equal to the divergence of the diffusive flux plus time rates of production and/or
destruction in chemical reactions. As d
V
ϒ
=(
ρ
w
,
ρ
1
,...,
ρ
N
)
→
0, mass balances of each polymeric con-
